THE STRUCTURE OF NASH EQUILIBRIA IN POISSON GAMES∗ CLAUDIA MERONI† AND CARLOS PIMIENTA‡ ABSTRACT. We show that many results on the structure and stability of equi- libria in finite games extend to Poisson games. In particular, the set of Nash equilibria of a Poisson game consists of finitely many connected components and at least one of them contains a stable set (De Sinopoli et al. [1]). In a simi- lar vein, we prove that the number of Nash equilibria in Poisson voting games under plurality, negative plurality, and (when there are at most three candi- dates) approval rule, as well as in Poisson coordination games, is generically finite. As in finite games, these results are obtained exploiting the geometric structure of the set of Nash equilibria which, in the case of Poisson games, is shown to be semianalytic. KEY WORDS. Poisson games, voting, stable sets, generic determinacy of equi- libria, o-minimal structures. JEL CLASSIFICATION. C70, C72. 1. INTRODUCTION Games with population uncertainty and, in particular, Poisson games (My- erson [2]), have been proposed to model economic scenarios where it is more reasonable to assume that agents only have probabilistic information about the number of players. Poisson games have been primarily used to study voting games (for a very incomplete list, see Myerson [3], Bouton and Castanheira [4], Bouton and Gratton [5], Bouton [6], Gratton [7], Huges [8]) but they also have been proven useful in more general economic environments where the number of economic agents is uncertain (see, e.g., Satterthwaite and Shneyerov [9], Makris [10, 11], Ritzberger [12], McLennan [13], Jehiel and Lamy [14]). Similarly to finite games, Poisson games can generate multiple equilibrium out- comes and, moreover, some of them may be only generated by equilibria that are not a plausible description of rational behavior. The economic literature has typ- ically addressed the issue of multiplicity of equilibrium outcomes investigating conditions that guarantee they are at least locally unique (Arrow and Hahn [15, † DEPARTMENT OF ECONOMICS,UNIVERSITY OF VERONA, VERONA,ITALY. ‡ SCHOOL OF ECONOMICS,THE UNIVERSITY OF NEW SOUTH WALES,SYDNEY, AUSTRALIA. Email addresses: [email protected], [email protected]. Corresponding author: Carlos Pimienta. Date: February 15, 2017. 1 2 CLAUDIA MERONI AND CARLOS PIMIENTA Chapter 9]). On the other hand, the literature on equilibrium refinements has helped reducing the indeterminacy of game-theoretical models by eliminating implausible equilibria. In this paper, we expand the analysis of the determinacy of equilibria to Poisson games. Local uniqueness of equilibrium outcomes in a theoretical model is a basic requirement from an applied standpoint. Otherwise, the theory would be too ambiguous in its description of what to expect from the economic agents. Fur- thermore, without such a property when performing comparative statics, small variations in the environment can lead to abrupt changes in behavior and in the ensuing outcomes. Local uniqueness of equilibrium outcomes is equivalent to finiteness if the strategy space is compact, which is typically the case. De- breu [16] argues that even if it is too demanding a requirement that a model has finitely many equilibrium outcomes for every possible description of the en- vironment, in some cases, it is possible to show that “most” of them satisfy such a finiteness result. Harsanyi [17] initiated the analysis of the determinacy of equilibrium in game theory and proved that, for generic assignments of utili- ties to strategy profiles, every normal form game has finitely many equilibria. This result is of limited significance because in many game-theoretical mod- els (such as voting games) many different strategy profiles lead to the same outcome and, therefore, lead to non-generic games in the space of normal form payoffs. Henceforth, the literature has provided analogous results for extensive- form games (Kreps and Wilson [18]), sender-receiver games (Park [19]), voting games (De Sinopoli [20]), network formation games (Pimienta [21]) and other families of finite games (Govindan and McLennan [22]). We show how many of these results can be extended to Poisson games. In particular, Poisson voting games under plurality, negative plurality, and approval rules (the latter when the number of candidates is no larger than three), as well as Poisson coordi- nation games, have finitely many Nash equilibria for generic utilities over the relevant outcome space. Generic determinacy results also have a practical implication for strategic stability, which has proved a powerful tool to characterize rational behavior in economic models. In finite games, the set of Nash equilibria consists of finitely many connected components. This is a key result towards a satisfactory theory of equilibrium refinements that conceives a solution to a game as a set-valued object (Kohlberg and Mertens [23], Mertens [24]). If there are finitely many equilibrium outcomes then every point in an equilibrium component necessarily induces the same outcome. In these cases, a set-valued solution concept con- tained in a connected component of equilibria constitutes a minor departure THE STRUCTURE OF NASH EQUILIBRIA IN POISSON GAMES 3 U(nc,σ) U(c,σ) σ(c) 0 σ∗ 1 FIGURE 1. The two Nash equilibria of the contribution game. from a classical single-valued one, as all the points in the same connected com- ponent can be considered equivalent. Motivated by this, in Section 3 we prove that every Poisson game has finitely many Nash equilibrium components and that at least one of them contains a stable set. Stable sets in Poisson games are introduced in De Sinopoli et al. [1] where it is shown that they satisfy existence, admissibility, and robustness against elim- ination of dominated actions and inferior replies. A stable set in a Poisson game is defined as a set of equilibria that is robust against a suitably chosen family of perturbations. A perturbation in this family is not a strategy tremble as in Kohlberg and Mertens [23], but rather, consists of pushing with vanishing prob- ability the population’s average behavior towards a completely mixed measure on the probability simplex over the set of actions in the game. [1] show that such perturbations generate associated utility-perturbed Poisson games. These per- turbed games are obtained by adding, for each action available to each type, a constant payoff to the original utility of playing such an action. This constant is equal to a vanishing fraction of the integral of the expected utility function asso- ciated with that action with respect to the completely mixed probability measure of the perturbation (see the Appendix for the formal definition.) To get a feeling for the definition of stability in Poisson games and its impli- cations, consider a simple contribution to a public good game. Suppose that the expected number of players is n > 0, that each player is endowed with one dollar, and that players can either commit to contribute (c) or not contribute (nc) to the provision of a public good that costs k dollars, where k < n. If less than k players are willing to contribute then the public good is not provided and no player pays anything. If k or more players are willing to contribute then the public good is provided and each of those players contributes equally to the cost. In terms of preferences, let us assume that players derive a utility from the public good that is sufficiently higher than the utility they derive from the dollar. 4 CLAUDIA MERONI AND CARLOS PIMIENTA Figure 1 represents a player’s expected utilities under the two actions as func- tions of the probability σ(c) that the average member of the population chooses to contribute (recall that, in a Poisson game, players of the same type play the same randomization over actions and that this game has only one type). Note that no action is dominated. If nobody is willing to contribute then either ac- tion leads to the same outcome and the player keeps her dollar anyway. On the other hand, if every player in the population is willing to contribute then the public good is provided with high enough probability so that the player prefers to free ride and not contribute to the public good. This game has two equilib- ria, an inefficient equilibrium in which no player contributes to the public good and an efficient equilibrium where players commit to contribute with probabil- ity σ∗ and are indifferent between contributing or not to the public good. The first equilibrium is not stable because it is not robust against every perturba- tion. In particular, take a perturbation whose probability measure assigns most probability to the segment of the simplex where the utility to contributing is larger than the utility to not contributing. Under this measure, the expected utility function of contributing integrates strictly more than the expected utility function of not doing so. Hence, the associated utility-perturbed game adds a constant to the utility of the first action strictly larger than the constant added to the utility of the second. And this perturbed game has no equilibrium close to the equilibrium of the true game where no player contributes. We conclude that the unique stable set of this game consists of the efficient equilibrium σ∗. In finite games, most game-theoretical constructions, such as the set of Nash equilibria, generate semialgebraic sets, i.e., sets defined by finite systems of poly- nomial inequalities. This implies that the set of Nash equilibria of every game consists of finitely many connected components and at least one of them con- tains a stable set (Kohlberg and Mertens [23]). Moreover, powerful results of semialgebraic geometry such as the Generic Local Triviality Theorem (Hardt [25], Bochnak et al. [26]) are used to prove most of the generic determinacy re- sults mentioned above.